Question: Find the greatest value of $t$ such that  \[\frac{t^2 - t -56}{t-8} = \frac{3}{t+5}.\]
We could cross-multiply, but that doesn't look like much fun.  Instead, we first factor the quadratic, which gives us  \[\frac{(t-8)(t+7)}{t-8} = \frac{3}{t+5}.\]Canceling the common factor on the left gives  \[t+7 = \frac{3}{t+5}.\]Multiplying both sides by $t+5$ gives us $(t+7)(t+5) = 3$.   Expanding the product on the left gives $t^2 + 12t + 35 = 3$, and rearranging this equation gives $t^2 +12 t + 32 = 0$.  Factoring gives $(t+4)(t+8) = 0$, which has solutions $t=-4$ and $t=-8$.  The greatest of these solutions is $\boxed{-4}$.